Haboush's theorem
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In mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple algebraic group G over a field K, and for any linear representation ρ of G on a K-vector space V, given v in V that is fixed by the action of G, there is a G-invariant F on V that is a homogeneous polynomial on V such that
- F(v) ≠ 0.
Here, more formally, we can take the symmetric powers of ρ to describe F. When K has characteristic 0 this was well known; the question was the extension to prime characteristic p. The conjecture was proved in 1975 by W. J. Haboush, about a decade after the problem had been posed by David Mumford, in the introduction to his book Geometric Invariant Theory.
It is of interest in relation to Mumford's geometric invariant theory, as well as some questions arising in classical parts of invariant theory. These with its help can be shown not to depend in an essential way on the characteristic of K.
There are other Mumford conjectures, for example on the mapping class group.
[edit] Reference
- W. J. Haboush, Reductive groups are geometrically reductive, Ann. of Math. (102) 1975, 67-83
